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The following results in the field Euclidean geometry, set theory, linear programming, nonstandard analysis, topology and number theory have to be called sensational! Because of the finiteness of our world, there are certain difficulties to treat the infinite. One difficulty is the question whether the number of elements of the set of algebraic numbers should be defined finite or infinite.

The finite definition offers significant advantages of handling and is traditional. It is shown that the sets of natural, integer, rational, algebraic, real or complex numbers are not closed. Hence, the difference of algebraic numbers is no longer necessarily algebraic, what complicates the theory of transcendental numbers. An infinite rational and transcendental number can also consist of a finite continued fraction.

Here the last partial denominator can be infinitely big. If we would identify it with a conventionally rational number by setting the last partial fraction equal to zero, it would simultaneously solve a linear equation with (infinite) integer coefficients. This identification leads to contradictions, if the first equation is subtracted several times from the second one and the solution of every newly emerged equation is determined.

It is correct to work with approximate fractions. We can show that, by (conventionally natural and infinite natural) induction that, starting with the set of conventionally natural numbers, it can be diagonalised up to any power according to Cantor, so that all(!) infinite sets are equipotent to the set of conventionally natural numbers, if we use Hilbert's translations as an aid.

This contradictoriness is met with the theorem that there is for no set a bijection to its proper subset. Hence, Dedekind-infinity and Hilbert's hotel are refuted, since the image sets of translations of every set lead out of the latter. We can specify every number of elements of a set by reference to the set of conventionally natural numbers. We obtain this only by precisely prescribing the exact construction.

It is something complete different whether we consider all multiples of five and thereby construct the associated set in such a way that each conventionally natural number is multiplied by five, or whether we delete all numbers, up to each fifth, from the set of the conventionally natural numbers. Cantor regarded such sets as of the same cardinal number. If we, however, consider bijections correctly, results another picture.

Cantor's distinction between merely countable and uncountable sets is too undifferentiated. The correct treatment of bijections yields the statement that, concerning the number, there are infinite many sets between the set of conventionally natural numbers and the one of conventionally real numbers. Thus, the continuum hypothesis gets a new answer. Furthermore, the asymptotic function of the number of algebraic numbers is determined.

Since individual n-ness belongs to every natural number n that cannot be derived from its predecessors or successors, there is no complete system of axioms in mathematics, because with each new number something irreducible new emerges. By confining, however, to selected aspects, we can specify a finite system of axioms for a finite number of entities. Each level of infinity refuses completeness all the more.

Mathematics is not value-free. Theories are based on presuppositions. In mathematics, they are often expressed by axioms (prove to be true or false resp. to justify). Thus, all theories are incomplete and, as the case may be, beyond that, contradictory. Instead of explicit axioms, (implicit) definitions are more suitable in that the existence of the specified is tacitly presupposed with all emerging difficulties, until refutation.

The Euclidean geometry gives three definitions that challenge several axioms, using results of set theory. The question of a fair distribution of persons deals less about the theoretical content than about practical application. Here a spread-sheet analysis is used. The set theory defines some new sets and states generally their number, especially for the algebraic numbers.

In the nonstandard analysis, integration, differentiation, continuity, convergence and limit value for finite and infinite (conventionally not measurable) sets are redefined (maybe by considering their homogeneity) and examples are given, as well as for discontinuous functions, to obtain better, more precise, more elegant or simply correct mathematical statements (also for known theorems of analysis).

The advice is the result of my mathematical experience and discusses appropriate ways and procedures for solving mathematical problems. In topology, the measure problem is solved, openness and closure of sets are reduced to absurdity as well as the real and complex numbers are stated as only Kolmogorov spaces resp. their Hausdorff property is denied.

The linear programming proves first the diameter theorem for polytopes. The known exponential simplex method with a new perturbation method is confronted with the polynomial intex method, which also solves linear systems. On the internet, a linear programme can be entered as a table in one area, which is solved in a second area. A second solution will be returned if it exists.

In number theory, necessary and sufficient criteria for transcendental numbers are stated with some new examples. The conjecture by Littlewood is refuted and conventionally proven. The conjecture by Alaoglu and Erdős as well as the generalised Riemann hypothesis are elementarily proven. The latter implies a prime number theorem being more precise and the ternary Goldbach theorem, as well as numerous further theorems.

In the calculation of times, it is shown how the octal system can be used practically worldwide and what the advantages are here. Besides introducing a new calendar and a new calculation of clock time, the octal system is applied also to the SI-units metre (m) and second (s). It is reasoned with practical examples why the octal system can completely replace the decimal system advantageously.

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